2021/08/02 by David Barrera, Barrera, David, Emmanuel Gobet +1
Engineering · Mathematics · #37A25 #62G08 #62R07 #Control Systems and Identification #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (stat.ML) #Markov Chains and Monte Carlo Methods #Statistical Methods and Inference #Statistics Theory (math.ST)
paper · doi:10.48550/arxiv.2108.00997
openalex publication_date 2021/08/02 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
In this paper we present a series of results that permit to extend in a direct manner uniform deviation inequalities of the empirical process from the independent to the dependent case characterizing the additional error in terms of β-mixing coefficients associated to the training sample. We then apply these results to some previously obtained inequalities for independent samples associated to the deviation of the least-squared error in nonparametric regression to derive corresponding generalization bounds for regression schemes in which the training sample may not be independent. These results provide a framework to analyze the error associated to regression schemes whose training sample comes from a large class of β-mixing sequences, including geometrically ergodic Markov samples, using only the independent case. More generally, they permit a meaningful extension of the Vapnik-Chervonenkis and similar theories for independent training samples to this class of β-mixing samples.